Efficient Monte Carlo Methods: Control Variate for Stochastic Differential Equations, Multiscale and Constrained Dynamics

We propose efficient Monte Carlo (MC) methods for a broad class of stochastic differential equations (SDEs). These equations can have multiscale properties or constrained solutions. The proposed methods will be part of the next generation of MC techniques and will provide effective tools to meet fundamental needs arising from scientific, industrial, or societal challenges. Indeed, experts from various disciplines need to model and simulate quantities formulated in terms of the aforementioned stochastic models. We will consider variance reduction strategies of MC estimators such as control variates, ad-hoc approaches, and their applications in numerical analysis for SDEs, finance, and materials science. We will introduce a control variate estimator for quantities expressed as the expectation of a function of a diffusion process. The control variate is built with the same function and with a random process that satisfies the same equation as the diffusion process except that the Brownian motion is replaced by its optimal polynomial approximation. This will yield a practical control variate estimator with a dramatic small variance. We will tackle the parameter calibration problem for observed option prices with a multiscale volatility model. We will estimate the cost functional of this optimization problem with a martingale control variate strategy. We will propose efficient Monte Carlo simulations for statistics of hysteretic elastoplastic systems subjected to colored noise via a control variate estimator driven by approximation diffusion. We will introduce a new long excursion formulation to characterize and simulate efficiently the diffusivity properties related to dry friction with noise as experimentally studied by physicists.